## The Fibonacci Sequence

The sequence \(1, 1, 2, 3, 5, \ldots\) is called the Fibonacci Sequence. It is the sequence \(f(1)\), \(f(2)\), \(f(3)\), \(f(4)\), \(\ldots\) defined by

- \(f(1) = 1\),
- \(f(2) = 1\), and
- \(f(n + 2) = f(n + 1) + f(n)\) for all \(n \ge 1\).

So for example \(f(12) = 144\). So the first two digits of \(f(12)\) are \(14\) and the last two digits of \(f(12)\) are \(44\).

Part A | What are the first three digits of \(f(123)\)? | |
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Part B | What are the first six digits of \(f(123456)\)? | |

Part C | What are the last six digits of \(f(123456)\)? | |

Part D | What are the first nine digits of \(f(123456789)\)? | |

Part E | What are the last nine digits of \(f(123456789)\)? |

## Digital Self Powers

The *self-power* of a number \(x\) is defined to be \(x^x\). The *digital-self-power* of a number is defined to be the sum of the self powers of all of its digits (in base 10). For example, the digital-self-power of \(402\) is \(4^4 + 0^0 + 2^2 = 256 + 1 + 4 = 261\).

Part A | Find an integer which is one more than its digital-self-power. | |
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Part B | Find an integer (greater than 1) which equals its digital-self-power | |

Part C | Find a positive integer which is one less than its digital-self-power | |

Part D | Find the largest integer which is at most one away from its digital-self-power |